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Metamath Proof Explorer

Theorem cbvmo

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvmow , cbvmovw when possible. (Contributed by NM, 9-Mar-1995) (Revised by Andrew Salmon, 8-Jun-2011) (Proof shortened by Wolf Lammen, 4-Jan-2023) (New usage is discouraged.)

Ref Expression
Hypotheses cbvmo.1 ⊢ Ⅎ y φ
cbvmo.2 ⊢ Ⅎ x ψ
cbvmo.3 ⊢ x = y → φ ↔ ψ
Assertion cbvmo ⊢ ∃* x φ ↔ ∃* y ψ

Proof

Step Hyp Ref Expression
1 cbvmo.1 ⊢ Ⅎ y φ
2 cbvmo.2 ⊢ Ⅎ x ψ
3 cbvmo.3 ⊢ x = y → φ ↔ ψ
4 1 sb8mo ⊢ ∃* x φ ↔ ∃* y y x φ
5 2 3 sbie ⊢ y x φ ↔ ψ
6 5 mobii ⊢ ∃* y y x φ ↔ ∃* y ψ
7 4 6 bitri ⊢ ∃* x φ ↔ ∃* y ψ